arXiv:1704.05321v5 [math.OC] 16 Sep 2018
A characterization of the Logarithmic Least Squares Method
L´aszl´o Csat´o
∗Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI) Laboratory on Engineering and Management Intelligence, Research Group of Operations
Research and Decision Systems Corvinus University of Budapest (BCE)
Department of Operations Research and Actuarial Sciences Budapest, Hungary
18th September 2018
Abstract
We provide an axiomatic characterization of the Logarithmic Least Squares Method (sometimes called row geometric mean), used for deriving a preference vector from a pairwise comparison matrix. This procedure is shown to be the only one satisfying two properties, correctness in the consistent case, which requires the reproduction of the inducing vector for any consistent matrix, and invariance to a specific transformation on a triad, that is, the weight vector is not influenced by an arbitrary multiplication of matrix elements along a 3-cycle by a positive scalar.
Keywords: Decision analysis; pairwise comparisons; geometric mean; axiomatic ap- proach; characterization
JEL classification number: C44 MSC class: 90B50, 91B08
1 Introduction
Pairwise comparisons are a fundamental tool in many decision-analysis methods such as the Analytic Hierarchy Process (AHP) (Saaty,1980). However, in real-world applications the judgements of decision-makers may be inconsistent: for example, alternative Ais two times better than alternative B, alternative B is three times better than alternative C, but alternative A is not six times better than alternativeC. Inconsistency can also be an inherent feature of the data, for example, on the field of sports (Csat´o,2013;Boz´oki et al., 2016; Chao et al.,2018).
Therefore, a lot of methods have been suggested in the literature for deriving prefer- ence values from pairwise comparison matrices. In such cases, it seems to be fruitful to
∗ e-mail: laszlo.csato@uni-corvinus.hu
follow an axiomatic approach: the introduction and justification of reasonable properties may help to narrow the set of appropriate weighting methods and reveal some crucial fea- tures of them. The most important contribution of similar analyses can be an axiomatic characterization, when a set of propertiesuniquely determine a preference vector.
Characterization of different methods is a standard tool in social choice theory, for instance, in the case of the Shapley value in game theory (see e.g. Shapley (1953);
van den Brink and Pint´er(2015)), or for the Hirsch index in scientometrics (see e.g. Woeginger (2008); Bouyssou and Marchant (2014)). This approach has been applied recently for in-
consistency indices of pairwise comparison matrices (Csat´o,2018a,b).
Fichtner (1984), presumably the first work on the axiomatizations of weighting meth-
ods, characterized the Logarithmic Least Squares Method (Rabinowitz,1976;Crawford and Williams, 1980,1985;De Graan,1980) by using four requirements, correctness in the consistent case,
comparison order invariance, smoothness, and power invariance. Fichtner (1986) showed that substituting power invariance with rank preservation leads to the Eigenvector Method suggested by Saaty (1980).
From this set of axioms, correctness in the consistent case and comparison order invari- ance are almost impossible to debate. However, according to Bryson(1995), there exists a goal-programming method satisfying power invariance and a slightly modified version of smoothness, which possesses the additional property that the presence of a single outlier cannot prevent the identification of the correct priority vector. While Fichtner (1984) introduces smoothness in terms of differentiable functions and continuous derivat- ives, the interpretation of Bryson (1995) – a small change in the input does not lead to a large change in the output – seems to be more natural for us. Cook and Kress (1988) approached the problem by focusing on distance measures in order to get another goal programming method on an axiomatic basis.
Smoothness and power invariance can be entirely left out from the characterization of the Logarithmic Least Squares Method. Barzilai et al. (1987) exchange them for a consistency-like axiom by considering two procedures: (1) some pairwise comparison matrices are aggregated to one matrix and the solution is computed for this matrix;
(2) the priorities are derived separately for each matrix and combined by the geometric mean; which are required to result in the same preference vector. We think it is not a simple condition immediately to adopt. Barzilai (1997) managed to replace this axiom and comparison order invariance with essentially demanding that each individual weight is a function of the entries in the corresponding row of the pairwise comparison matrix only. Joining to Dijkstra (2013), we are also somewhat uncomfortable with this premise.
To summarize, the problem of weight derivation does not seem to be finally settled by the axiomatic approach. Consequently, it may not be futile to provide another character- ization of the Logarithmic Least Squares Method, which hopefully highlights some new aspects of the procedure. This is the main aim of the current paper.
Presenting an axiomatic characterization does not mean that we accept all properties involved as wholly justified and unquestionable or we reject the axioms proposed by previ- ous works. To consider an example from a related topic, although most axiomatic analysis of inconsistency (Brunelli and Fedrizzi, 2011; Brunelli, 2016, 2017; Brunelli and Fedrizzi,
2015,2018;Cavallo and D’Apuzzo,2012;Koczkodaj and Szwarc,2014;Koczkodaj and Szybowski, 2015;Koczkodaj and Urban,2018) look for well-motivated axioms that should be satisfied
by any reasonable measure, Csat´o(2018b) does not deal with the appropriate motivation of his axioms, the issue to be investigated is only how they can narrow the set of incon- sistency indices.
This paper strictly follows the latter direction, therefore, we only say thatif one agrees with our axioms, then the geometric mean remains the only choice. The importance of an axiomatic characterization does not depend on a convincing explanation for the de- sirability of the properties suggested. For example, Arrow’s impossibility theorem (Arrow, 1950) can also be interpreted as an axiomatization of the dictatorial rule by universality, Pareto efficiency, and independence of irrelevant alternatives, however, it does not mean that the dictatorial rule is good.
The study is structured as follows. Section 2presents some definitions on the field of pairwise comparison matrices. Two properties of weighting methods are defined in Sec- tion 3, which will provide the characterization of the Logarithmic Least Squares Method in Section 4. Section 5 summarizes our findings.
2 Preliminaries
Assume that n alternatives should be measured with respect to a given criterion on the basis of pairwise comparisons such that ai,j is an assessment of the relative importance of alternative i with respect to alternative j.
Let Rn+ and Rn+×n denote the set of positive (with all elements greater than zero) vectors of size n and matrices of size n×n, respectively.
Definition 2.1. Pairwise comparison matrix: Matrix A = [ai,j] ∈ Rn+×n is a pairwise comparison matrix if aj,i = 1/ai,j for all 1≤i, j ≤n.
Any pairwise comparison matrix is well-defined by its elements above the diagonal since we discuss only multiplicative pairwise comparison matrices with the reciprocal property throughout the paper. Let An×n be the set of pairwise comparison matrices of size n×n.
A pairwise comparison matrix A ∈ An×n is called consistent if ai,k = ai,jaj,k for all 1 ≤i, j, k ≤ n. Otherwise, it is said to be inconsistent. Any pairwise comparison matrix is allowed to be inconsistent unless its consistency is explicitly stated.
Definition 2.2. Weight vector: Vector w= [wi]∈Rn+ is a weight vector if Pni=1wi = 1.
LetRn be the set of weight vectors of size n.
Definition 2.3. Weighting method: Function f :An×n→ Rn is a weighting method.
A weighting method associates a weight vector to any pairwise comparison matrixA such that fi(A) is the weight of alternative i.
Several weighting methods have been suggested in the literature, seeChoo and Wedley (2004) for an overview. This paper discusses two of them, which are among the most
popular.
Definition 2.4. Eigenvector Method (EM) (Saaty, 1980): The Eigenvector Method is the function A →wEM(A) such that
AwEM(A) =λmaxwEM(A) and
n
X
i=1
wEMi = 1,
whereλmaxdenotes the maximal eigenvalue, also known as principal or Perron eigenvalue, of matrix A.
Definition 2.5. Logarithmic Least Squares Method (LLSM) (Crawford and Williams, 1980, 1985; De Graan, 1980): The Logarithmic Least Squares Method is the function A → wLLSM(A) such that the weight vector wLLSM(A) is the optimal solution of the problem:
w∈Rminn
n
X
i=1 n
X
j=1
"
logai,j−log wi
wj
!#2
. (1)
LLSM is sometimes called (row) geometric mean because the solution of (1) can be computed as
wLLSMi (A) =
Qn j=1a1/ni,j
Pn
k=1
Qn
j=1a1/nk,j . (2)
3 Axioms
In this section, two properties of weighting methods will be discussed.
Axiom 3.1. Correctness (CO): Let A ∈ An×n be a consistent pairwise comparison matrix. Weighting method f : An×n → Rn is correct if fi(A)/fj(A) = ai,j for all 1≤i, j ≤n.
CO requires the reproduction of the inducing vector for any consistent pairwise com- parison matrix. It was introduced by Fichtner (1984) under the name correct result in the consistent case and was used by Fichtner (1986), Barzilai et al. (1987) and Barzilai (1997), among others.
Proposition 3.1. The Eigenvector Method and the Logarithmic Least Squares Method satisfy correctness.
Definition 3.1. α-transformation on a triad: Let A ∈ An×n be a pairwise comparison matrix and 1 ≤ i, j, k ≤ n be three different alternatives. An α-transformation on the triad (i, j, k) – which is determined by the three alternatives i, j, and k – provides the pairwise comparison matrix ˆA ∈ An×n such that α > 0, ˆai,j = αai,j (ˆaj,i = aj,i/α), ˆ
aj,k = αaj,k (ˆak,j = ak,j/α), ˆak,i = αak,i (ˆai,k = ai,k/α) and ˆaℓ,m = aℓ,m for all other elements.
The transformation changes three elements of a pairwise comparison matrix along a 3-cycle. It can reproduce local consistency: the choice α = q3 ai,k/(ai,jaj,k) leads to ˆ
ai,jˆaj,k =α2ai,jaj,k =ai,k/α= ˆai,k.
Naturally, this process modifies all values of the triad, while maybe two of the compar- isons are accurate and one contains all the inaccuracy. However, if no further information is available, then our assumption seems to be reasonable.
Axiom 3.2. Invariance to α-transformation on a triad (IT): Let A,Aˆ ∈ An×n be any two pairwise comparison matrices such that ˆA can be obtained from A through an α-transformation on a triad. Weighting method f : An×n → Rn is invariant to α- transformation on a triad if f(A) =f( ˆA).
IT means that the weights of the alternatives are not influenced byα-transformations on triads. It has been inspired by the axiomindependence of circuits inBouyssou (1992).
A motivation for invariance toα-transformation on a triad can be the following. Con- sider a sports competition where player i has defeated player j, player j has defeated player k, while player k has defeated player i, and suppose that the three wins are equi- valent. Then the final ranking is not allowed to change if the margins of victories are modified by the same amount. In particular, the three results can be reversed (j beats i, k beats j, and i beats k), or all comparisons can become a draw. IT is practically a generalization of this idea.
Proposition 3.2. The Eigenvector Method violates invariance to α-transformation on a triad.
Proof. Consider the following pairwise comparison matrices:
A=
1 1 1 8
1 1 1 1
1 1 1 1
1/8 1 1 1
and Aˆ =
1 2 1 4
1/2 1 1 2
1 1 1 1
1/4 1/2 1 1
.
Aˆ can be obtained fromA through an α-transformation on the triad (1,2,4) by α= 2 as ˆ
a1,2 = 2a1,2, ˆa1,4 =a1,4/2, and ˆa2,4 = 2a2,4. The corresponding weight vectors are wEM(A) ≈ h 0.4269 0.2182 0.2182 0.1367 i⊤ 6=
6= h 0.3941 0.2256 0.2389 0.1413 i⊤ ≈ wEM( ˆA), which shows the violation of the axiom IT.
Proposition 3.3. The Logarithmic Least Squares Method satisfies invariance toα-transformation on a triad.
Proof. Take two pairwise comparison matrices A,Aˆ ∈ An×n such that ˆA is obtained from A through an α-transformation on a triad, namely, they are identical except for ˆ
ai,j = αai,j (ˆaj,i = aj,i/α), ˆaj,k = αaj,k (ˆak,j = ak,j/α), and ˆak,i = αak,i (ˆai,k = ai,k/α).
The product of row elements does not change, so wLLSM(A) = wLLSM( ˆA) according to (2).
Corollary 3.1. The counterexample concerning the Eigenvector Method and IT in Pro- position 3.2 is minimal with respect to the number of alternatives as in the case of n= 3, EM and LLSM yield the same result (Crawford and Williams, 1985).
4 Characterization of the Logarithmic Least Squares Method
Theorem 4.1. The Logarithmic Least Squares Method is the unique weighting method satisfying correctness and invariance to α-transformation on a triad.
Proof. LLSM satisfies both axioms according to Propositions 3.1 and 3.3.
For uniqueness, consider an arbitrary pairwise comparison matrix A ∈ An×n and a weighting method f : An×n → Rn, which meets correctness and invariance to α- transformation on a triad. Denote byPi = qnQnk=1ai,kthe geometric mean of row elements
for alternative i. In order to prove thatf is equivalent to the Logarithmic Least Squares Method, it is enough to show that fi(A)/fj(A) =Pi/Pj.
The proof will present that every pairwise comparison matrix can be transformed into a consistent matrix by a sequence of α-transformations on a triad carried out over an appropriately chosen set of triads: the triads to be modified always contain the first al- ternative, while the two others are considered according to the sequence (n−1, n),(n− 2, n),(n−2, n−1),(n−3, n),(n−3, n−1),. . . , (2, n),(2, n−1),. . . ,(2,3). Since the Log- arithmic Least Squares Method satisfies invariance to α-transformation on a triad, each matrix of the sequence shares the same geometric mean weight vector. The procedure will stop after (maximally) (n−1)(n−2)/2 steps, which is the number of triads not containing the first alternative.
Let us introduce the pairwise comparison matrixA(n−1,n) ∈ An×nsuch thata(n1,n−−1,n)1 :=
αn−1,nan−1,n; a(n1,n−1,n) := a1,n/αn−1,n; a(nn−−1,n1,n) := αn−1,na1,n1 and ai,j := a(ni,j−1,n) for all other elements, where αn−1,n = Pn−1/(Pnan−1,n). Since A(n−1,n) is obtained from A through anα-transformation on a triad,f(A) =fA(n−1,n)according to the assumption that f satisfies the axiom IT. If n = 3, A(n−1,n) is consistent because a(2,3)1,2 = P1/P2, a(2,3)1,3 =P1/P3, and a(2,3)2,3 =P2/P3, therefore correctness implies f(A) =wLLSM(A).
Otherwise, analogous α-transformations on a triad can be implemented until we get A(i,j)∈ An×n, where 1< i < j and a(i,j)k,ℓ =Pk/Pℓ for all i≤ k < ℓ. The next step of the algorithm depends on the difference of i and j.
• If j > i+ 1, then introduce the pairwise comparison matrix A(i,j−1) ∈ An×n such that a(i,j1,i−1) :=αi,j−1a(i,j)1,i ; a(i,j1,j−1) :=a(i,j)1,j /αi,j−1; a(i,ji,j−−11) := αi,j−1a(i,j)i,j−1 and a(i,jk,ℓ−1) :=a(i,j)k,ℓ for all other elements, where αi,j−1 =Pi/Pj−1a(i,j)i,j−1
. It can be checked that a(i,jk,ℓ−1) =a(i,j)k,ℓ =Pk/Pℓ for all i≤k < ℓ, whilea(i,ji,j−−11) =Pi/Pj−1.
• Ifj =i+1 andi >2, then define the pairwise comparison matrixA(i−1,n)∈ An×n such that a(i1,i−−1,n)1 := αi−1,na(i,n)1,i−1; a(i1,n−1,n) := a(i,n)1,n /αi−1,n; a(ii−−1,n1,n) := αi−1,na(i,n)i−1,n
and a(i,jk,ℓ−1) := a(i,j)k,ℓ for all other elements, where αi−1,n = Pi−1/Pna(i,n)i−1,n
. It can be checked that a(ik,ℓ−1,n) = a(i,n)k,ℓ = Pk/Pℓ for all i ≤ k < ℓ, while a(ii−−1,n1,n) = Pi−1/Pn.
Finally, A(2,3) ∈ An×nis obtained such that a(2,3)k,ℓ =Pk/Pℓ for all 2≤k < ℓ. Furthermore, a(2,3)1,j =a1,j
Qn
m=j+1αj,m
Qj−1 m=2αm,j
=a1,j
n
Y
m=j+1
Pj
Pm 1 aj,m
j−1
Y
m=2
Pj
Pmam,j
.2
However, am,j = 1/aj,m due to the reciprocity condition and Qnm=1aj,m =Pjn, therefore a(2,3)1,j = Pjn−2
Qn
m=j+1PmQj−1 m=2Pm
1 Pjn = 1
Pj
1
Qn m=2Pm
.
It is clear that P1 = 1/(Qnm=2Pm) as the product of all elements of A gives one, which leads to
a(2,3)1,j = P1 Pj
1 For the sake of simplicity, only the elements above the diagonal are indicated.
2 Note thatQj−1
α = 1 ifj = 2 andQn
α = 1 ifj=n.
for all j ≥ 2. In other words, A(2,3) ∈ An×n is a consistent pairwise comparison mat- rix such that a(2,3)i,j = Pi/Pj = wLLSMi A(2,3)/wLLSMj A(2,3) for all 1 ≤ i, j ≤ n.
Consequently, fA(2,3) = wLLSMA(2,3) due to correctness. Weighting method f is invariant to α-transformation on a triad, hence
fA(2,3)=fA(2,4)=· · ·=fA(n−1,n)=f(A).
The Logarithmic Least Squares Method also satisfies IT according to Proposition 3.3, verifying the claim that f(A) =wLLSM(A).
Example 4.1. As an illustration of the proof of Theorem 4.1, consider the following pairwise comparison matrix:
A =
1 1 1 16
1 1 1 1
1 1 1 1
1/16 1 1 1
, which leads to wLLSM(A) = 1 9
4 2 2 1
.
Since a3,4 6= w3LLSM(A)/w4LLSM(A), an α-transformation on the triad (1,3,4) should be carried out by α3,4 = hwLLSM3 (A)/w4LLSM(A)i/a3,4 = 2, which results in the matrix A(3,4). After that, another α-transformation on the triad (1,2,4) is necessary by α2,4 =
hw3LLSM(A)/wLLSM4 (A)i/a(3,4)2,4 = 2 in order to get the matrixA(2,4):
A(3,4) =
1 1 2 8
1 1 1 1
1/2 1 1 2
1/8 1 1/2 1
and A(2,4) =
1 2 2 4
1/2 1 1 2
1/2 1 1 2
1/4 1/2 1/2 1
=A(2,3).
Finally, anα-transformation on the triad (1,2,3) should be carried out byα2,3 =hwLLSM2 (A)/wLLSM3 (A)i/a(3,4)2,3 = 1, so the pairwise comparison matrix remains unchanged, A(2,3) = A(2,4). It is a con-
sistent matrix, therefore any weighting method satisfying correctness and invariance to α-transformation on a triad should give wLLSM(A) as the weight vector associated with the pairwise comparison matrix A.
Proposition 4.1. CO and IT are logically independent axioms.
Proof. It is shown that there exist weighting methods, which satisfy one axiom, but do not meet the other:
1 CO: the Eigenvector Method (see Propositions 3.1 and 3.2);
2 IT: the flat method such that fi(A) = 1/n for all 1≤i≤n.
5 Conclusions
We have proved LLSM to be the unique weighting method among the procedures used to derive priorities from reciprocal pairwise comparison matrices, which is correct in the consistent case and invariant to a specific transformation on a triad. The somewhat
surprising fact is that our algorithm aims only to recover local consistency by focusing on a given triad without the consideration of other elements of the pairwise comparison matrix. Hence satisfaction of a local property fully determines a global weight vector.
Naturally, one can debate whether the axiom invariance toα-transformation on a triad should be accepted, but, at least, it reveals an important aspect of the geometric mean, contributing to the long list of its favourable theoretical properties (Barzilai et al., 1987;
Barzilai, 1997; Dijkstra, 2013; Caklovi´c and Kurdija,ˇ 2017; Lundy et al., 2017; Csat´o, 2018c). Furthermore, the violation of this property can be an argument against the Eigen- vector Method, a procedure having several other disadvantages, for example, the Pareto inefficiency of the weight vector (Blanquero et al.,2006; Boz´oki,2014; Boz´oki and F¨ul¨op, 2018), or the possibility of strong rank reversal in group decision-making (P´erez and Mokotoff, 2016; Csat´o,2017).
Some directions for future research are also worth mentioning. First, further axiomatic analysis and characterizations of weighting methods can help in a better understanding of them. Second,α-transformation on a triad seems to be related to inconsistency reduction processes in pairwise comparison matrices (Koczkodaj and Szybowski, 2016; Szybowski, 2018). Third, the Logarithmic Least Squares Methods has been extended to the in- complete case when certain elements of the pairwise comparison matrix are unknown (Boz´oki et al.,2010). Axiomatization on this more general domain seems to be promising and within reach, as revealed byBoz´oki and Tsyganok(2017), althoughLLSMsometimes behaves strangely on this general domain (Csat´o and R´onyai, 2016).
Acknowledgements
We would like to thank Denis Bouyssou for inspiration and S´andor Boz´oki for plentiful advice.
Three anonymous reviewers provided beneficial remarks and suggestions.
We are grateful to the audience of our talk at Corvinus Game Theory Seminar on 17 March 2017 for useful comments concerning the proof of Theorem 4.1.
The research was supported by OTKA grant K 111797 and by the MTA Premium Post Doctorate Research Program.
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